SparseMultivariateSkewPolynomial(R, Var, sigma, delta)ΒΆ

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SparseMultivariateSkewPolynomial(R, Var, sigma, delta) defines a mutivariate Ore ring over R in variables from V. sigma(v) gives automorphism of R corresponding to variable v and delta(v) gives corresponding derivative.

0: %
from AbelianMonoid
1: %
from MagmaWithUnit
*: (%, %) -> %
from Magma
*: (%, Fraction Integer) -> % if R has Algebra Fraction Integer
from RightModule Fraction Integer
*: (%, R) -> %
from RightModule R
*: (Fraction Integer, %) -> % if R has Algebra Fraction Integer
from LeftModule Fraction Integer
*: (Integer, %) -> %
from AbelianGroup
*: (NonNegativeInteger, %) -> %
from AbelianMonoid
*: (PositiveInteger, %) -> %
from AbelianSemiGroup
*: (R, %) -> %
from LeftModule R
+: (%, %) -> %
from AbelianSemiGroup
-: % -> %
from AbelianGroup
-: (%, %) -> %
from AbelianGroup
/: (%, R) -> % if R has Field
from AbelianMonoidRing(R, IndexedExponents Var)
=: (%, %) -> Boolean
from BasicType
^: (%, NonNegativeInteger) -> %
from MagmaWithUnit
^: (%, PositiveInteger) -> %
from Magma
~=: (%, %) -> Boolean
from BasicType
annihilate?: (%, %) -> Boolean
from Rng
antiCommutator: (%, %) -> %
from NonAssociativeSemiRng
associates?: (%, %) -> Boolean if R has EntireRing
from EntireRing
associator: (%, %, %) -> %
from NonAssociativeRng
binomThmExpt: (%, %, NonNegativeInteger) -> % if % has CommutativeRing
from FiniteAbelianMonoidRing(R, IndexedExponents Var)
characteristic: () -> NonNegativeInteger
from NonAssociativeRing
charthRoot: % -> Union(%, failed) if R has CharacteristicNonZero
from CharacteristicNonZero
coefficient: (%, IndexedExponents Var) -> R
from AbelianMonoidRing(R, IndexedExponents Var)
coefficient: (%, List Var, List NonNegativeInteger) -> %
from MaybeSkewPolynomialCategory(R, IndexedExponents Var, Var)
coefficient: (%, Var, NonNegativeInteger) -> %
from MaybeSkewPolynomialCategory(R, IndexedExponents Var, Var)
coefficients: % -> List R
from FiniteAbelianMonoidRing(R, IndexedExponents Var)
coerce: % -> % if R has CommutativeRing and % has VariablesCommuteWithCoefficients or R has IntegralDomain and % has VariablesCommuteWithCoefficients
from Algebra %
coerce: % -> OutputForm
from CoercibleTo OutputForm
coerce: Fraction Integer -> % if R has Algebra Fraction Integer or R has RetractableTo Fraction Integer
from Algebra Fraction Integer
coerce: Integer -> %
from NonAssociativeRing
coerce: R -> %
from Algebra R
commutator: (%, %) -> %
from NonAssociativeRng
content: % -> R if R has GcdDomain
from FiniteAbelianMonoidRing(R, IndexedExponents Var)
D: Var -> %
D(v) returns operator corresponding to derivative with respect to v in R.
degree: % -> IndexedExponents Var
from AbelianMonoidRing(R, IndexedExponents Var)
degree: (%, List Var) -> List NonNegativeInteger
from MaybeSkewPolynomialCategory(R, IndexedExponents Var, Var)
degree: (%, Var) -> NonNegativeInteger
from MaybeSkewPolynomialCategory(R, IndexedExponents Var, Var)
Delta: Symbol -> % if Var has variable: Symbol -> Var
Delta(s) returns operator corresponding to derivative with respect to s in R.
exquo: (%, %) -> Union(%, failed) if R has EntireRing
from EntireRing
exquo: (%, R) -> Union(%, failed) if R has EntireRing
from FiniteAbelianMonoidRing(R, IndexedExponents Var)
fmecg: (%, IndexedExponents Var, R, %) -> %
from FiniteAbelianMonoidRing(R, IndexedExponents Var)
ground: % -> R
from FiniteAbelianMonoidRing(R, IndexedExponents Var)
ground?: % -> Boolean
from FiniteAbelianMonoidRing(R, IndexedExponents Var)
hash: % -> SingleInteger
from SetCategory
hashUpdate!: (HashState, %) -> HashState
from SetCategory
latex: % -> String
from SetCategory
leadingCoefficient: % -> R
from AbelianMonoidRing(R, IndexedExponents Var)
leadingMonomial: % -> %
from AbelianMonoidRing(R, IndexedExponents Var)
leftPower: (%, NonNegativeInteger) -> %
from MagmaWithUnit
leftPower: (%, PositiveInteger) -> %
from Magma
leftRecip: % -> Union(%, failed)
from MagmaWithUnit
mainVariable: % -> Union(Var, failed)
from MaybeSkewPolynomialCategory(R, IndexedExponents Var, Var)
map: (R -> R, %) -> %
from AbelianMonoidRing(R, IndexedExponents Var)
mapExponents: (IndexedExponents Var -> IndexedExponents Var, %) -> %
from FiniteAbelianMonoidRing(R, IndexedExponents Var)
minimumDegree: % -> IndexedExponents Var
from FiniteAbelianMonoidRing(R, IndexedExponents Var)
monomial: (%, List Var, List NonNegativeInteger) -> %
from MaybeSkewPolynomialCategory(R, IndexedExponents Var, Var)
monomial: (%, Var, NonNegativeInteger) -> %
from MaybeSkewPolynomialCategory(R, IndexedExponents Var, Var)
monomial: (R, IndexedExponents Var) -> %
from AbelianMonoidRing(R, IndexedExponents Var)
monomial?: % -> Boolean
from AbelianMonoidRing(R, IndexedExponents Var)
monomials: % -> List %
from MaybeSkewPolynomialCategory(R, IndexedExponents Var, Var)
numberOfMonomials: % -> NonNegativeInteger
from FiniteAbelianMonoidRing(R, IndexedExponents Var)
one?: % -> Boolean
from MagmaWithUnit
opposite?: (%, %) -> Boolean
from AbelianMonoid
pomopo!: (%, R, IndexedExponents Var, %) -> %
from FiniteAbelianMonoidRing(R, IndexedExponents Var)
primitiveMonomials: % -> List %
from MaybeSkewPolynomialCategory(R, IndexedExponents Var, Var)
primitivePart: % -> % if R has GcdDomain
from FiniteAbelianMonoidRing(R, IndexedExponents Var)
recip: % -> Union(%, failed)
from MagmaWithUnit
reducedSystem: (Matrix %, Vector %) -> Record(mat: Matrix Integer, vec: Vector Integer) if R has LinearlyExplicitOver Integer
from LinearlyExplicitOver Integer
reducedSystem: (Matrix %, Vector %) -> Record(mat: Matrix R, vec: Vector R)
from LinearlyExplicitOver R
reducedSystem: Matrix % -> Matrix Integer if R has LinearlyExplicitOver Integer
from LinearlyExplicitOver Integer
reducedSystem: Matrix % -> Matrix R
from LinearlyExplicitOver R
reductum: % -> %
from AbelianMonoidRing(R, IndexedExponents Var)
retract: % -> Fraction Integer if R has RetractableTo Fraction Integer
from RetractableTo Fraction Integer
retract: % -> Integer if R has RetractableTo Integer
from RetractableTo Integer
retract: % -> R
from RetractableTo R
retractIfCan: % -> Union(Fraction Integer, failed) if R has RetractableTo Fraction Integer
from RetractableTo Fraction Integer
retractIfCan: % -> Union(Integer, failed) if R has RetractableTo Integer
from RetractableTo Integer
retractIfCan: % -> Union(R, failed)
from RetractableTo R
rightPower: (%, NonNegativeInteger) -> %
from MagmaWithUnit
rightPower: (%, PositiveInteger) -> %
from Magma
rightRecip: % -> Union(%, failed)
from MagmaWithUnit
sample: %
from AbelianMonoid
smaller?: (%, %) -> Boolean if R has Comparable
from Comparable
subtractIfCan: (%, %) -> Union(%, failed)
from CancellationAbelianMonoid
totalDegree: % -> NonNegativeInteger
from MaybeSkewPolynomialCategory(R, IndexedExponents Var, Var)
totalDegree: (%, List Var) -> NonNegativeInteger
from MaybeSkewPolynomialCategory(R, IndexedExponents Var, Var)
totalDegreeSorted: (%, List Var) -> NonNegativeInteger
from MaybeSkewPolynomialCategory(R, IndexedExponents Var, Var)
unit?: % -> Boolean if R has EntireRing
from EntireRing
unitCanonical: % -> % if R has EntireRing
from EntireRing
unitNormal: % -> Record(unit: %, canonical: %, associate: %) if R has EntireRing
from EntireRing
variables: % -> List Var
from MaybeSkewPolynomialCategory(R, IndexedExponents Var, Var)
zero?: % -> Boolean
from AbelianMonoid

AbelianGroup

AbelianMonoid

AbelianMonoidRing(R, IndexedExponents Var)

AbelianSemiGroup

Algebra % if R has CommutativeRing and % has VariablesCommuteWithCoefficients or R has IntegralDomain and % has VariablesCommuteWithCoefficients

Algebra Fraction Integer if R has Algebra Fraction Integer

Algebra R if R has CommutativeRing and % has VariablesCommuteWithCoefficients

BasicType

BiModule(%, %)

BiModule(Fraction Integer, Fraction Integer) if R has Algebra Fraction Integer

BiModule(R, R)

CancellationAbelianMonoid

canonicalUnitNormal if R has canonicalUnitNormal

CharacteristicNonZero if R has CharacteristicNonZero

CharacteristicZero if R has CharacteristicZero

CoercibleTo OutputForm

CommutativeRing if R has IntegralDomain and % has VariablesCommuteWithCoefficients or R has CommutativeRing and % has VariablesCommuteWithCoefficients

CommutativeStar if R has CommutativeRing and % has VariablesCommuteWithCoefficients or R has IntegralDomain and % has VariablesCommuteWithCoefficients

Comparable if R has Comparable

EntireRing if R has EntireRing

FiniteAbelianMonoidRing(R, IndexedExponents Var)

FullyLinearlyExplicitOver R

FullyRetractableTo R

IntegralDomain if R has IntegralDomain and % has VariablesCommuteWithCoefficients

LeftModule %

LeftModule Fraction Integer if R has Algebra Fraction Integer

LeftModule R

LinearlyExplicitOver Integer if R has LinearlyExplicitOver Integer

LinearlyExplicitOver R

Magma

MagmaWithUnit

MaybeSkewPolynomialCategory(R, IndexedExponents Var, Var)

Module % if R has CommutativeRing and % has VariablesCommuteWithCoefficients or R has IntegralDomain and % has VariablesCommuteWithCoefficients

Module Fraction Integer if R has Algebra Fraction Integer

Module R if R has CommutativeRing and % has VariablesCommuteWithCoefficients

Monoid

MultivariateSkewPolynomialCategory(R, IndexedExponents Var, Var)

NonAssociativeRing

NonAssociativeRng

NonAssociativeSemiRing

NonAssociativeSemiRng

noZeroDivisors if R has EntireRing

RetractableTo Fraction Integer if R has RetractableTo Fraction Integer

RetractableTo Integer if R has RetractableTo Integer

RetractableTo R

RightModule %

RightModule Fraction Integer if R has Algebra Fraction Integer

RightModule R

Ring

Rng

SemiGroup

SemiRing

SemiRng

SetCategory

unitsKnown